Question

question 5 (10 points) given ( moverarc{be} = 128^circ ) and ( mangle bfe = 46^circ ), determine the measure of the arc ( moverarc{cd} ). (you may assume that point a is the center of the circle.) a) b) c) d)

Explanation:

Step1: Recall the secant - angle formula

The measure of an angle formed by two secants intersecting outside a circle is equal to half the difference of the measures of the intercepted arcs. The formula is \(m\angle BFE=\frac{1}{2}(m\widehat{BE}-m\widehat{CD})\).

Step2: Substitute the known values

We know that \(m\angle BFE = 46^{\circ}\) and \(m\widehat{BE}=128^{\circ}\). Substitute these values into the formula:
\(46^{\circ}=\frac{1}{2}(128^{\circ}-m\widehat{CD})\)

Step3: Solve for \(m\widehat{CD}\)

First, multiply both sides of the equation by 2:
\(2\times46^{\circ}=128^{\circ}-m\widehat{CD}\)
\(92^{\circ}=128^{\circ}-m\widehat{CD}\)

Then, rearrange the equation to solve for \(m\widehat{CD}\):
\(m\widehat{CD}=128^{\circ}- 92^{\circ}\)
\(m\widehat{CD}=36^{\circ}\)

Answer:

\(36^{\circ}\) (assuming the option with \(36^{\circ}\) is the correct one, for example, if option c is \(36^{\circ}\), then the answer is c) \(36^{\circ}\))